Lines through segments in 3D space

Given a set of n line segments in three-dimensional space, finding all the lines that simultaneously intersect at least four line segments in is a fundamental problem that arises in a variety of domains. We refer to this problem as the lines-through-segments problem, or LTS for short. We present an efficient output-sensitive algorithm and its implementation to solve the LTS problem. The implementation is exact and properly handles all degenerate cases. To the best of our knowledge, this is the first implementation for the LTS problem that is (i) output sensitive and (ii) handles all degenerate cases. The algorithm runs in O((n ³ + I)logn) time, where I is the output size, and requires O(nlogn + J) working space, where J is the maximum number of output elements that intersect two fixed line segments; I and J are bounded by O(n ⁴) and O(n ²), respectively. We use Cgal arrangements and in particular its support for two-dimensional arrangements in the plane and on the sphere in our implementation. The efficiency of our implementation stems in part from careful crafting of the algebraic tools needed in the computation. We also report on the performance of our algorithm and its implementation compared to others. The source code of the LTS program as well as the input examples for the experiments can be obtained from http://acg.cs.tau.ac.il/ projects/lts.